Center for TurbulenceResearch Proceedings of the Summer Program Anisotropic eddy viscosity mo dels By D Carati AND W Cab ot A general discussion on the structure of the eddy viscosity tensor in anisotropic ows is presented The systematic use of tensor symmetries and ow symmetries is shown to reduce drastically the numb er of indep endent parameters needed to describ e the rank eddy viscosity tensor The p ossibility of using Onsager symmetries for simplifying further the eddy viscosity is discussed explicitly for the axisymmetric geometry Intro duction Contrary to most of the works presented in this volume this note do es not re sult from a planned pro ject for the summer program It develop ed instead from discussions during the course of the workshop bymany participants concerning the representation of anisotropy in the mo deling of the subgridscale stress in Large Eddy Simulation LES This study is thus an attempt to present a systematic dis cussion of the inuence of anisotropy on the structure of the eddy viscosity tensor Some of the results presented here are not really original since they have b een de rived in other contexts visco elastic media or magnetized plasmas However we found several motivations for repro ducing the general study of tensor symmetries in the sp ecial case of the eddy viscosity tensor First we remark that there is often evidence of anisotropy at the subgrid level The most obvious case arises when the grid itself is anisotropic In that case even if the ow do es satisfy the classical lo cal isotropy assumption the subgrid velo city would b e anisotropic by construction Since most LESs use a nonuniform grid with anisotropic stretching the eects of anisotropy should b e taken into account in a very wide class of problems Second the discussions we had during the workshop showed that few attempts have b een made to intro duce the anisotropy at the tensor level in the relation b e tween the subgrid scale stress and the resolved strain tensor On the contrary most of the studies on the inuence of anisotropyhave fo cused on p ossible mo dications to the isotropic eddy viscosity amplitude Deardor Scotti et al Finally the development of the dynamic pro cedure Germano Ghosal et al Lilly allows the intro duction of multiparameter mo dels for the subgrid scale stress Therefore there is no practical reason for practitioners to limit their mo dels to an isotropic eddy viscosity Universite Libre de Bruxelles NASA Center for Turbulence Research D Carati W Cabot Anisotropic eddy viscosity In this work we only consider the subgrid scale mo deling of an incompressible uid If the exact description of the large scale pressure is not required the trace of the subgrid scale tensor may b e added to the pressure which is then calculated in order to ensure the incompressibility The only tensor that needs to b e mo deled is u u u u u u u u i j i j k k k k ij ij The usual mo deling pro cedure consists in giving an expression for in terms of the ij u These quantities are usually spatial derivatives of the resolved velo city eld j i decomp osed into a symmetric resolved strain tensor S u u ij j j i i and an antisymmetric resolved rotation tensor R u u w ij i j j i ij k k where w is the vorticity and is the LeviCivita fully antisymmetric tensor with k ij k The most general tensorial relation in an anisotropic system thus reads S R ij k l kl ij k l kl ij For three dimensional turbulence a naive analysis of this relation would lead to the conclusion that b oth and are describ ed by indep endent parameters However very strong simplications can b e derived by using the tensor symmetry properties of S and R aswell as the symmetries of the ow These simplications do not ij ij ij require any assumption as far as the mo del is accepted A more debatable simplication might apply if the Onsager reciproca l symmetries Onsager are assumed to hold for the eddy viscosity tensors This will b e discussed at the end of this section Tensor symmetries The tensors and S are symmetric and traceless while the tensor R is an ij ij ij tisymmetric This implies that the eddy viscosity tensor has the following ij k l prop erties ij k l jikl ij k l ij lk iik l ij k k Thus for a given value of k lk l the matrix a is traceless and ij ij k l symmetric Consequently it has indep endent comp onents Similarlyfora given Anisotropic eddy viscosity models value of i j i j the matrix b is also traceless and symmetric The kl i j kl full tensor is thus describ ed by indep endent parameters The same ij k l analysis can b e p erformed for the tensor which has the following symmetries ij k l ij k l jikl ij k l ij lk iik l Now the tensor is symmetric and traceless for its rst two indices while ij k l it is antisymmetric for its last two indices Consequently the full tensor is ij k l describ ed by indep endent parameters Flow symmetries This parameter eddy viscosity tensor may b e strongly simplied byusing the symmetries of the ow Let us consider some simple cases Isotropic turbulence Any isotropic tensor can only b e constructed with the unit tensor Thus the ij most general isotropic tensor of rank can b e written as follows T a a a ij k l ij kl ik jl il jk If we imp ose the symmetry relations it turns out that the eddy viscosity tensor reduces to a ij k l ik jl il jk ij kl while the symmetry relations imply that the tensor vanishes Consequently the subgrid scale stress reads a S ij ij where a is the usual isotropic eddy viscosity Smagorinsky The simplest anisotropic situation arises when only one direction can b e distin guished from the other This axisymmetric geometry is thus characterized bya vector p ointing to the anisotropy direction We will show that the nature of this vector will strongly aect the structure of the eddy viscosity tensor In particular anisotropy induced by a pseudovector like a magnetic eld or a rotation must b e treated dierently from the anisotropy induced by an axial vector likeamean ow Axisymmetry induced by an axial vector We rst consider the case of an axisymmetry characterized by an axial vector n An axisymmetric tensor of rank can only b e a function of this vector n and of the unit tensor Its most general form compatible with the symmetry b etween the ij rst two indices reads D Carati W Cabot b b T ij kl ik jl il jk ij k l b n n b n n b n n n n ij k l i j kl ik j l jk i l b n n n n b n n n n il j k jl i k i j k l Imp osing the constraints and dening b c b c and b c lead to the following expressions b c c n c n b b c c n b c If the constraints are imp osed on only two parameters are dierent ij k l from zero and are opp osite b b Thus byintro ducing b c in the subgridscale stress reads c S c n s s n s n ij i j i j ij k k ij n s n c r n n r c n n ij k k i j i j i j s S n and r R n The eect on the resolved energy balance of the where i ik k i ik k rst three terms is fully determined by the sign of the parameters c c andc Indeed these terms corresp ond to dissipation resp creation of resolved energy if and only if c c andc are p ositive resp negative On the contrary the sign of the term prop ortional to c in the resolved energy balance dep ends simultaneously on the sign of c and on the ow through the factor s r k k S c jS j c s c s n c s r ij k k k k ij If the anisotropyisweak n is relatively small only terms up to n must b e retained since s r O n the term prop ortional to c can b e neglected in this case i i Axisymmetry induced by a pseudovector Wenow consider that the anisotropy direction is represented by a pseudovector p The most general axisymmetric tensor of rank will b e a function of the vector p the unit tensor and the LeviCivita tensor The situation is thus more i ij ij k complicated and more parameters need to b e intro duced The notations will b e simplied byintro ducing the antisymmetric tensor V p so that the most ij ij k k general tensor compatible with the symmetry b etween the rst two indices reads T d d d V d V V ij kl ik jl il jk ij kl ik jl jk il ij k l d V V d p p d p p d p p il jk jl ik ik l j jkl i ij k l i j kl a d p p p p d p p p p ik j l jk i l il j k jl i k d V V V V d V p p V p p ik jl il jk ik j l jk i l d V p p V p p d p p V d p p p p il j k jl i k i j kl i j k l Anisotropic eddy viscosity models We will not discuss the complete tensor T with indep endent parameters Let us assume that the anisotropyisweak enough to keep only terms prop ortional to the vector p In this case T reduces to i T d d ij kl ik jl il jk ij k l b d V d V V ij kl ik jl jk il d V V d p p il jk jl ik ik l j jkl i Imp osing the constraints and dening d e d e and d e lead to the following expression d e d e d e e while imp osing the constraints with the new denition d e and d e leads to d d d e e d e The subgrid scale stress thus reads e S e e S V S V ij ik jk jk ik ij a R V R V R V e e ij ik jk jk ik kl kl Although the total eddy viscosity contains parameters only three of them app ear indep endently in the expression for Let us note that the expression of can ij ij b e simplied by using the resolved vorticity S e e S V S V e ij ik